# One-Plane Glide-Symmetric Holey Structures for Stop-Band and Refraction Index Reconfiguration

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{Air}and the dielectric substrate h

_{Diel}, together with the dielectric constant ε

_{r}of the substrate and the period p. Periodic conditions in the x- and y-directions of the unit cell were applied to perform the dispersion analysis. The direction of propagation selected in the analysis extends along the y-axis, in which there are two periodicities to generate glide symmetry. In the perpendicular direction (x), there is only one periodicity. Figure 1b shows the parameters related to the size, position, orientation, and shape of the ellipses, which will determine the basic behavior of the whole structure. In particular, the sizes of the major and minor semi-axes of the ellipses are defined by the values of A

_{1}and B

_{1}for the first ellipse and A

_{2}and B

_{2}for the second ellipse. The parameters φ and θ define the angle of inclination of the first and second ellipses. A zero value means a horizontal position of the ellipses (semi-minor axis in the y-direction). Positive values of φ introduce counter-clockwise rotation for the first ellipse, and positive values of θ imply clockwise rotation for the second ellipse. Finally, the distance between ellipses can be modified by the displacement parameter d. The ellipses are kept at a distance equal to the periodicity p with d = 0, while values greater than zero mean that the second ellipse moved toward the first ellipse.

_{1}= A

_{2}, B

_{1}= B

_{2}, φ = θ, and d = 0. The symmetry axis is at the center of the structure along the y-direction, and glide symmetry is obtained by reflecting the initial ellipse through the symmetry axis and displacing the ellipse obtained at a distance p in the y-direction. Figure 2 depicts an example of the operation of this structure. Figure 2a represents the dispersion diagrams of a structure with glide symmetry, and Figure 2b represents the same structure after breaking the symmetry with a modification of θ. In this work, all the dispersion analyses were carried out using the Eigenmode solver in CST Microwave Studio considering periodic boundaries in x- and y-directions, and electrical boundaries in the z-direction. Although there are available methods to quickly analyze glide symmetry based on the Floquet theorem [12,13,14] or equivalent circuits [11,29], these methods were not proposed to analyze this structure [30]. The dispersion diagrams only represent phase variations in the y-direction, as it is the direction in which the glide symmetry takes place. The chosen parameters for the case shown in Figure 2a are as follows: ε

_{r}= 3, h

_{Air}= 0.3 mm, h

_{Diel}= 1.52 mm, p = 2.75 mm, A

_{1}= A

_{2}= 2.5 mm, B

_{1}= B

_{2}= 0.75 mm, φ = θ = 30°, and d = 0 mm. The value of θ is changed to 90° in the structure of Figure 2b.

_{v}, and frequency in the medium under study, f

_{m}, for any β

_{v}(2p)/π = β

_{m}(2p)/π value of the dispersion diagram, assuming that periodicity p is the same in both cases. Since β

_{v}(2p)/π = β

_{m}(2p)/π, we have λ

_{v}= λ

_{m}; thus, we get Equation (2).

_{r}= 3, h

_{Air}= 0.3 mm, h

_{Diel}= 1.52 mm, p = 2.75 mm, A

_{1}= A

_{2}= 2.5 mm, B

_{1}= B

_{2}= 0.75 mm, φ = θ = 30°, and d = 0 mm). This section is subdivided into four main subsections, whereby each represents an alternative to break the symmetry and open stop-bands. Rupture of the symmetry is achieved by varying four fundamental parametric relationships: the size ratio between ellipses A

_{2}/A

_{1}, the vertical displacement of the second ellipse normalized with respect to the period d/p, the orientation of the ellipses φ and θ, and the relationship between semi-minor axis B

_{1}

_{,}B

_{2}and semi-major axis A

_{1}

_{,}A

_{2}of the ellipses. The graphs also include other parameters that do not play a role in breaking the symmetry, but that can intensify its effect. At the end of this section, the effect of varying the period p is presented, for glide and non-glide configurations. All graphs represent the reference points at the right side of the dispersion diagram (β(2p)/π = 1), as indicated in Figure 2. The continuous lines indicate the lower limit of the stop-band, while the dashed lines indicate the upper limit.

#### 2.1. Symmetry Broken by the Ellipse Size

_{1}/A

_{1}= B

_{2}/A

_{2}= 0.3). In Figure 3, the stop-band opens when the second ellipse reduces its size, A

_{2}, to near zero while keeping the size A

_{1}constant (A

_{2}/A

_{1}= 0.1). It should be noted that the stop-band width begins to reach saturation for values less than A

_{2}/A

_{1}= 0.5. This means that it is not necessary to reach very small ellipse sizes to achieve a larger stop-band, which facilitates the manufacturing. As expected, for an A

_{2}/A

_{1}= 1 ratio, the glide symmetry is recovered and the stop-band is completely closed. Additionally, results are included for different sizes of the first ellipse (A

_{1}from 1.5 mm to 2.5 mm). The value of the period p of the structure was kept constant at 2.75 mm. It can be seen how the reduction in size of this ellipse leads to narrower stop-bands. In addition, the refractive index of the structure increases as A

_{1}increases. This is consistent, as the fields propagating in the air gap are most affected by the dielectric for larger ellipse sizes.

#### 2.2. Symmetry Broken by the Displacement between Ellipses

_{Air}/p. The value of p is 2.75 mm for all the cases. The rupture of the symmetry is greater upon increasing d/p values; thus, the stop-band increases. Of special interest is the significant effect that reducing the thickness of the air gap has on the stop-band widening. The explanation for this phenomenon is once again that the effect of the elliptical structures intensifies when the field is more confined within the air gap. This also explains the shifting to lower frequencies for smaller thicknesses as the refractive index increases.

#### 2.3. Symmetry Broken by the Orientation of the Ellipses

_{Diel}/$\sqrt{{\epsilon}_{r}}$ (ε

_{r}= 3). It can be observed that the curves cross each other when φ and θ are equal, which happens in cases where glide symmetry exists. Therefore, the stop-band opens for θ values that differ more from φ. The curves have a sinusoid shape associated with the rotation of the ellipses, for which only a half of the period is represented, since a similar behavior is expected in the other half (θ values from 90° to 180°). Regarding the thickness of the dielectric substrate, Figure 5 shows how the width of the stop-band increases for greater thicknesses, but this effect tends to saturate from a certain thickness. This is because the mode that penetrates the ellipses into the dielectric is an evanescent mode. Thus, when the thickness is small, the mode does not have enough space to be attenuated; however, for thicknesses greater than a certain threshold, the mode is attenuated enough so that it is not affected by an increase in thickness.

#### 2.4. Stop-Band Frequency Shifting and Semi-Minor vs. Semi-Major Axis Relationship

_{Air}= 0.3 mm, h

_{Diel}= 1.52 mm, A

_{1}= A

_{2}= 2.5∙(p/2.75) mm, and d = 0 mm. A parametric sweep of the B

_{1}and B

_{2}values was also included in the study, so that the ratio of the semi-minor and semi-major axes B/A (B = B

_{1}= B

_{2}; A = A

_{1}= A

_{2}) of the ellipses varied from 0.1 to 1. The results are shown in Figure 7a,b. Figure 7a contains the dispersion diagram value at β(2p)/π = 1 of a structure with glide symmetry (φ = 30°, θ = 30°). The analysis on a broken-symmetry structure (φ = 0°, θ = 90°) is depicted in Figure 7b.

#### 2.5. Regeneration of the Glide Symmetry

#### 2.6. Materials and Instrumentation

_{r}= 3.66; tan(δ) = 0.0035 @10GHz). The necessary aluminum parts were machined in an external company. Twenty 2.92-mm coaxial connectors with a core diameter of 0.3 mm and length of 7 mm were chosen for the connections. The calibration kit used was the 85056K HP/Agilent. The measurements were made using two 2.92-mm low-phase-error coaxial cables minibend KR-6, and eight loads Anritsu K210. Finally, the vector network analyzer used was the Agilent 8722ES model.

## 3. Results

_{r}= 3.66, h

_{Air}= 0.2 mm, h

_{Diel}= 1.52 mm, p = 2.75 mm, A

_{1}= A

_{2}= 2.5 mm, B

_{1}= B

_{2}= 0.75 mm, and d = 0 mm. The only values that were different were φ and θ, taking φ = θ = 30° for the glide configuration and φ = 0°, θ = 90° for the non-glide and regenerated configurations. The glide (Figure 12a) and non-glide (Figure 12b) designs contained 10 × 10 unit cells, whereas the design with regenerated symmetry (Figure 12c) contained only 5 × 10 unit cells due to the unit cell being twice as wide. The fourth design (Figure 12d) was a supporting structure that acted as a “thru” and that was used to make phase corrections in the measurements of the other three designs. The prototypes were excited by 10 coaxial connections, five on each side, to create a plane wave that emulated the boundary conditions used in the analyzed unit cells. It should be noted that we added a non-glide frame of ellipses around the four designs to prevent the propagation of spurious modes between the metal surface of the dielectric substrate and the aluminum casing that shielded these designs. As we can observe in Figure 12, the total length of the surface with ellipses was 55 mm (each unit cell was 5.5 mm in length), but the distance between the coaxial connectors and the ellipses was 24 mm in total. This is the separation that should exist between coaxial inputs and outputs of the “thru” for a correct post processing of the measurements.

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Unit cell of the proposed structure and parametrization. View in perspective (

**a**), and top view without the top metal plate (

**b**).

**Figure 2.**Dispersion diagrams for two configurations of the unit cell: glide-symmetric (

**a**) and non-glide-symmetric (

**b**).

**Figure 3.**Effect of the size of the second ellipse versus the first ellipse A

_{2}/A

_{1}for different values of A

_{1}. The rest of the parameters are ε

_{r}= 3, h

_{Air}= 0.3 mm, h

_{Diel}= 1.52 mm, p = 2.75 mm, B

_{1}/A

_{1}= B

_{2}/A

_{2}= 0.3, φ = θ = 30°, and d = 0 mm.

**Figure 4.**Effect of the displacement d of the second ellipse toward the first ellipse for different values of the air gap thickness h

_{Air}/p. The rest of the parameters are ε

_{r}= 3, h

_{Diel}= 1.52 mm, p = 2.75 mm, A

_{1}= A

_{2}= 2.5 mm, B

_{1}= B

_{2}= 0.75 mm, and φ = θ = 30°.

**Figure 5.**Effect of the orientation φ and θ of each of the two ellipses for different values of the dielectric substrate thickness h

_{Diel}/$\sqrt{{\epsilon}_{r}}$. The rest of the parameters are ε

_{r}= 3, h

_{Air}= 0.3 mm, p = 2.75 mm, A

_{1}= A

_{2}= 2.5 mm, B

_{1}= B

_{2}= 0.75 mm, and d = 0 mm.

**Figure 6.**Maximum fractional stop-band found for each value of φ in Figure 5 considering different substrate thicknesses.

**Figure 7.**Effect of the semi-minor vs. semi-major axis B/A relationship for different values of the period p: values for a glide-symmetric structure (

**a**) and a non-glide structure (

**b**). The rest of the parameters are h

_{Air}= 0.3 mm, A

_{1}= A

_{2}= 2.5∙(p/2.75) mm, and d = 0 mm.

**Figure 8.**Bandwidth of the stop-band for the non-glide configuration in Figure 7b as a function of B/A for different values of periodicity p.

**Figure 9.**The non-glide-symmetric structure in Figure 7b mirrored using a different symmetry axis (

**a**); the new glide-symmetric structure (

**b**) is obtained from (

**a**) after applying a translation.

**Figure 11.**Unit cells of the manufactured designs: glide-symmetric configuration (

**a**), non-glide configuration (

**b**), and regenerated glide configuration (

**c**).

**Figure 12.**Elliptic configurations of the final designs: glide configuration (

**a**), non-glide configuration (

**b**), regenerated glide configuration (

**c**), and “thru” section (

**d**).

**Figure 13.**Pieces of the metal casing for the “thru” section. Inner view of the top piece (

**a**); cross-section with the probes entering from the bottom part (

**b**); cross-section with the probes entering from the top part (

**c**); detailed view of the stepped transition (

**d**); details of the top piece (

**e**); details of bottom pieces (

**f**).

**Figure 14.**Pictures of the manufactured prototypes and assembly. Dielectric substrate with the elliptic patterns (

**a**); top and bottom parts of the casing (

**b**); detail of the transition at the top part of the casing (

**c**); final measurement assembly (

**d**).

**Figure 15.**Detailed pictures of the elliptic configurations: glide configuration (

**a**), non-glide configuration (

**b**), and regenerated glide configuration (

**c**).

**Figure 16.**Simulated and measured S parameters of the “thru” section (

**a**), the glide configuration (

**b**), the non-glide configuration (

**c**), and the regenerated glide configuration (

**d**).

**Figure 17.**Simulated and measured phases in transmission. Phase introduced by the “thru” section (

**a**), and dispersion diagrams of the glide (

**b**), non-glide (

**c**), and regenerated glide (

**d**) configurations.

**Figure 18.**Representations of the electric field inside the air gap of the simulated prototypes for the three configurations under study at given frequencies of interest.

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**MDPI and ACS Style**

Tamayo-Dominguez, A.; Fernandez-Gonzalez, J.-M.; Quevedo-Teruel, O.
One-Plane Glide-Symmetric Holey Structures for Stop-Band and Refraction Index Reconfiguration. *Symmetry* **2019**, *11*, 495.
https://doi.org/10.3390/sym11040495

**AMA Style**

Tamayo-Dominguez A, Fernandez-Gonzalez J-M, Quevedo-Teruel O.
One-Plane Glide-Symmetric Holey Structures for Stop-Band and Refraction Index Reconfiguration. *Symmetry*. 2019; 11(4):495.
https://doi.org/10.3390/sym11040495

**Chicago/Turabian Style**

Tamayo-Dominguez, Adrian, Jose-Manuel Fernandez-Gonzalez, and Oscar Quevedo-Teruel.
2019. "One-Plane Glide-Symmetric Holey Structures for Stop-Band and Refraction Index Reconfiguration" *Symmetry* 11, no. 4: 495.
https://doi.org/10.3390/sym11040495